2-coherent and 2-convex Conditional Lower Previsions
arXiv:1606.06043 · doi:10.1016/j.ijar.2016.06.003
Abstract
In this paper we explore relaxations of (Williams) coherent and convex conditional previsions that form the families of -coherent and -convex conditional previsions, at the varying of . We investigate which such previsions are the most general one may reasonably consider, suggesting (centered) -convex or, if positive homogeneity and conjugacy is needed, -coherent lower previsions. Basic properties of these previsions are studied. In particular, we prove that they satisfy the Generalized Bayes Rule and always have a -convex or, respectively, -coherent natural extension. The role of these extensions is analogous to that of the natural extension for coherent lower previsions. On the contrary, -convex and -coherent previsions with either are convex or coherent themselves or have no extension of the same type on large enough sets. Among the uncertainty concepts that can be modelled by -convexity, we discuss generalizations of capacities and niveloids to a conditional framework and show that the well-known risk measure Value-at-Risk only guarantees to be centered -convex. In the final part, we determine the rationality requirements of -convexity and -coherence from a desirability perspective, emphasising how they weaken those of (Williams) coherence.
This is the authors' version of a work that was accepted for publication in the International Journal of Approximate Reasoning, vol. 77, October 2016, pages 66-86, doi:10.1016/j.ijar.2016.06.003, http://www.sciencedirect.com/science/article/pii/S0888613X16300792